6533b7defe1ef96bd12769af
RESEARCH PRODUCT
The Riemannian manifold of all Riemannian metrics
Olga Gil-medranoPeter W. Michorsubject
Mathematics - Differential GeometryChristoffel symbolsGeneral MathematicsPrescribed scalar curvature problem58D17 58B20Mathematical analysisCurvatureLevi-Civita connectionFunctional Analysis (math.FA)Mathematics - Functional Analysissymbols.namesakeDifferential Geometry (math.DG)symbolsFOS: MathematicsSectional curvatureMathematics::Differential GeometryExponential map (Riemannian geometry)Ricci curvatureScalar curvatureMathematicsdescription
In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like curvature” and ”scalar like curvature”. The pointwise character mentioned above allows us in section 3, to solve explicitly the geodesic equation and to obtain the domain of definition of the
| year | journal | country | edition | language |
|---|---|---|---|---|
| 1991-12-31 |